How to select and move multiple vertices
There are multiple ways of doing this told below:
Box select: press the B key and you can click and drag a square that will select everything inside.

Circle select: press the C key, and your mouse will change to a circle, anything clicked inside the circle will be selected.

Holding shift: You can manually select vertices while holding Shift down to keep the previous selections as well. 

Select Linked: With one vertex selected of an object and pressing CtrlL you will select the whole object itself that is part of that one vertices you selected at the start.

It seems you are a beginner these option should be enough and to get you started, then you can use the transformation manipulators (using arrows on the screen) or press the G while selected to move it as well.